Perform the following operations and express your answer in the form a+b i. (13)/(5-12 i)

Name: Problem 21, Chapter 10: Higher Level Mathematics
Uploaded: 2022-01-19T18:44:04-08:00
Duration: 6 min 22 s
Channel: Asif Khan
Description: Perform the following operations and express your answer in the form a+b i. (13)/(5-12 i)

Perform the following operations and express your answer in...

Key Concepts

Division of Complex Numbers

This concept involves dividing one complex number by another. The process typically requires eliminating the complex number from the denominator by multiplying both numerator and denominator by the conjugate of the denominator. This yields an expression in the standard form a + bi.

Complex Conjugates

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. Multiplying a complex number by its conjugate results in a real number, as the imaginary parts cancel out. This property is crucial for simplifying complex fractions.

Standard Form of a Complex Number

The standard form for a complex number is expressed as a + bi, where a and b are real numbers and i represents the square root of -1. Presenting complex numbers in this form facilitates easier addition, subtraction, and interpretation of the real and imaginary components.

Multiplication of Complex Numbers

Multiplication of complex numbers involves applying the distributive property (often remembered as FOIL) and using the fact that i² = -1 to simplify expressions. This process is essential when multiplying a complex number by its conjugate or performing other operations involving complex numbers.

Transcript

00:01 Hi, today in this video we will learn the concept of dividing a real number by a complex number and we are given a question that is 13 divided by 5 minus 12 biota where 13 is a real number in numerator and you can see here 5 minus 12 in denominator is a complex number now let me tell you about a little bit about the complex number complex number is denoted by small z that is equals to a plus b aota this is the general form of the complex number where a is a real number and this part is called real part and b is imaginary number and a iota is the imaginary unit and this part is said to be the imaginary part and this whole is said to be the complex number now let me tell you about little bit the division if you divide like 13 by for example 26 real by real number if you divide i'm just giving you an example this will be 13 1 times 13 2 it will be 1 over 2 it means that dividing real number by a real number is very easy but to divide a real number by imaginary so a complex number that's a little bit difficult to make it easy i will do one thing in the denominator is we have a complex number i will convert this one into the real number and we know that for that we have we can do one thing because a plus b a is a complex number if we take in product like it's complex and you where that is a minus b aota we know that we get a square plus b square that is a real number so by by using this concept i will just do one thing to get the real number in the denominator i will multiply this denominator with its complex conjugate that is 5 plus 12 aorta but to keep the value of this fraction unchanged i will also multiply with the the complex conjugate with the numerator as well like 5 plus 12 aorta here as well now numerator will be multiplied with numerator and denominator will be denominator so 13 will be multiplied with 5 plus 12 aota and the in the denominator you can see you can observe one thing here if we if i take 5…

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